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Integraal van $$$\frac{x^{2}}{\sqrt{x^{21}}}$$$
Gerelateerde rekenmachine: Rekenmachine voor bepaalde en oneigenlijke integralen
Uw invoer
Bepaal $$$\int \frac{x^{2}}{\sqrt{x^{21}}}\, dx$$$.
Oplossing
De invoer is herschreven: $$$\int{\frac{x^{2}}{\sqrt{x^{21}}} d x}=\int{\frac{1}{x^{\frac{17}{2}}} d x}$$$.
Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=- \frac{17}{2}$$$:
$${\color{red}{\int{\frac{1}{x^{\frac{17}{2}}} d x}}}={\color{red}{\int{x^{- \frac{17}{2}} d x}}}={\color{red}{\frac{x^{- \frac{17}{2} + 1}}{- \frac{17}{2} + 1}}}={\color{red}{\left(- \frac{2 x^{- \frac{15}{2}}}{15}\right)}}={\color{red}{\left(- \frac{2}{15 x^{\frac{15}{2}}}\right)}}$$
Dus,
$$\int{\frac{1}{x^{\frac{17}{2}}} d x} = - \frac{2}{15 x^{\frac{15}{2}}}$$
Voeg de integratieconstante toe:
$$\int{\frac{1}{x^{\frac{17}{2}}} d x} = - \frac{2}{15 x^{\frac{15}{2}}}+C$$
Antwoord
$$$\int \frac{x^{2}}{\sqrt{x^{21}}}\, dx = - \frac{2}{15 x^{\frac{15}{2}}} + C$$$A